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Convective and Radiant Heat Transfer CHE 0201 Thursday A-5 Bridget Csongradi Karah Horbach Melissa Lane Catherine McElhinny Gabrielle Schantz Progress Report Spring 2015 Table of Contents Nomenclature .............................................................................................................. 2 1.0 Introduction and Background ...................................................................................... 2 2.0 Experimental Methodology ......................................................................................... 4 2.1 Equipment and Apparatus ................................................................................ 4 2.2 Experimental Procedures ................................................................................. 4 3.0 Results ........................................................................................................................ 5 4.0 Analysis and Discussion of Results ............................................................................. 6 5.0 Summary and Conclusions .......................................................................................... 6 6.0 Future Work ................................................................................................................ 7 7.0 References ................................................................................................................... 8 8.0 Appendix A-1.............................................................................................................. 9 9.0 Appendix A-2.............................................................................................................. 13 2 Progress Report Spring 2015 Nomenclature Symbol Definition Units As Surface Area m2 D Diameter of Heated Cylinder m F Area or geometric factor - hc Heat Transfer Coefficient, Natural Convection W/ m2k hf Heat Transfer Coefficient, Forced Convection W/ m2k hr Heat Transfer Coefficient, Radiation W/ m2k I Heater Current A k Thermal Conductivity W/mk Nu Nusselt Number - Pr Prandtl Number - Qc Heat Transfer, Natural Convection W Qf Heat Transfer, Forced Convection W Qr Heat Transfer, Radiation W Qtot Total Heat Loss W Re Reynolds Number - T1 Temperature of Surrounding Air °C/K T2 Temperature of Heater Surface °C/K Ua Air Velocity m/s Ue Effective Air Velocity m/s V Voltage to Heater V π Stefan Boltzmann Constant for Radiation - π Emissivity of cylinder - v Dynamic Viscosity of Air m2/s 3 Progress Report Spring 2015 1.0 Introduction and Background There are three main types of heat transfer: conduction, convection, and radiation. These types of heat transfers occur naturally and constantly throughout the universe and can be used in practical applications any time a room, object, or process requires heat. Conduction is the transfer of heat from a hot body to a cold body in which no material is flowing, usually through a solid. Convection involves the transfer of heat from one location to another by the movement of fluids, such as liquids or gases. The fluid carries energy with it as it moves from a location of higher temperature to one of lower temperature. Lastly, radiation is the transfer of heat by electromagnetic waves, or light. Thermal radiation is a type of heat transfer since the electromagnetic radiation emitted from a central source carries energy as the wave moves [1]. Convection is a type of heat transfer via moving fluids that can be utilized in process equipment. Depending on how the flow begins, the convection can be natural or forced. Natural convection is any fluid movement by natural means such as warmer fluid moving upward and cooler fluid moving downward. A type of driving force would also be a difference in density between two locations, resulting in the heat of one fluid being absorbed by another fluid. Natural convection can be found throughout nature, such as in earthβs oceans and atmosphere, which are heated by this force [1]. Forced convection occurs when a fluid flows over a surface by induced external forces, like a pump, fan, or mixer. The motion of the fluid increases heat transfer; there is a direct relationship between velocity and heat transfer- higher velocity equals more heat transfer [2]. A practical example of this type of heat transfer would be home heating systems which heat the air by force. Air in this equipment is heated by a type of furnace and blown by fans into a room. The fan acts as the driving force for the fluid to move into the room and transfer the heat gained by the furnace into the room [1]. Radiation can also be used in process equipment as a method of heat transfer of electromagnetic waves. Waves, which contain energy, are emitted from a source and move towards surrounding objects. These objects or surfaces absorb the energy possessed by the waves. This energy increases the average kinetic energy of the objectsβ particles, which raises the temperature of the object or surface [1]. 4 Progress Report Spring 2015 The Combined Convection and Radiation Heat Transfer Unit allows one to investigate both natural convection with radiation and forced convention. The surface temperature can be varied throughout the experiment in order to study the effect of increasing temperature on radiant heat transfer. Air velocity flowing through the cylinder on the equipment can also be varied to measure its effects on forced convection. This machine takes advantage of natural convection and radiation as well as forced convection, when the blower is turned on. First, radiation and heat loss by natural convection can be measured. By increasing the temperature of the equipment, radiation heats the air. Air transfers this heat to the environment as well as the surface of the equipment via natural convection as the heat moves from the hotter air to colder surfaces and fluids. Second, the equipment can be used to measure forced convection by varying the air velocity using an air blower, transferring the heat as the fluid is blown through the equipment. With radiation and natural convection, the fraction of heat loss from the heating element that occurs by radiation and natural convection to the air at steady-state can be calculated using experimental data found by testing the equipment. The proportion lost due to radiation depends on the surface temperature of the cylinder and the surroundings, the type of material of the cylinder, and the emissivity (efficiency with which a surface emits thermal energy) of the cylinder and the surroundings. The heat of natural convection can be calculated as follows: Qc=hcAs(Ts-Ta) (1) Where hc is the overall heat transfer coefficient due to natural convection, A s is the heated surface area of the cylinder, Ts is the temperature of the cylinder, and T a is the temperature of the surrounding air. The heat transfer of radiation is found using: Qr=hrAs(Ts-Ta) (2) Where hr is the overall heat transfer coefficient due to radiant convection. The total heat transfer from the cylinder is determined as follows: Qtot=Qr+Qc (3) For forced convection, the air velocityβs effect on the surface temperature as well as the rate of convective heat transfer can be studied from the heat transfer unit as well. 5 Progress Report Spring 2015 Similar to the previous equations, the heat loss due to forced convection can be found using: Qf=hfAs(Ts-Ta) (4) Combining this with the same heat transfer equation found for the radiant component gives an equation for total heat transfer for the forced convection analysis: Qtot=Qr+Qf (5) By comparing the relationship between the total heat transfer of forced convection as well as the surface temperature of the heating unit, the effect of air velocity on heat transfer can be determined. This experiment involved the testing of the radiant and natural convection heat transfer by varying the temperature and finding the amount of heat lost through convection and radiation. The second part of the experiment involved varying the air velocity into the cylinder as the air blower was on and analyzing the relationship between air velocity and forced convection heat transfer. 2.0 Experimental Methodology 2.1 Equipment and Apparatus β The Combined Convection and Radiation H112D heat transfer module (see Figures 1 & 2 below) was used in this experiment to determine the heat loss due to radiant heat transfer and natural convection of the air as well as the effect of air velocity on the heating element and the rate of convective heat transfer from the heating element to the air. This unit has two displays for temperatures T1 and T2; T2 is the heating surface temperature and T1 is the air temperature thermocouple. A knob is located on the unit adjusts the voltage and measures and displays the current in amperes. 6 Fig. 1. Combined Convection and Radiation H112D with its heat transfer module. Progress Report Spring 2015 The combined convection and radiation equipment is a tall cylindrical duct with an inlet to allow air to flow and also contains a heated cylinder at the top of the duct. The anemometer is attached to the middle of the duct and records the air velocity, which is displayed digitally at the bottom of the device. The air is brought through the fans and is controlled by a damper that determines how much space is open for the air to flow through. 2.2 Experimental Procedures β In order to start the procedure, the voltage Fig. 2. Labeled schematic of the Combined Convection and Radiation Unit must be turned to zero and the air flow must be fully opened. The initial air velocity was steadied at 0 m/s and the temperatures at T1 and T2 were at room temperature. While completing the first technical objective, the air blower remained off and the voltage dial was adjusted until the T2 temperature reached approximately 100°C. After this temperature reached steady state, when temperature and air velocity were constant for 5 seconds, measurements of the range of T1, T2, current, and voltage were recorded. T2 was increased in increments of approximately 100°C by increasing the voltage, and T1, T2, current, and voltage were recorded after reaching steady state. The final measurement was read at a temperature of 400°C. Technical objective two started with the air damper fully opened and the voltage set to approximately 200 V before adjusting the damper to allow for an air velocity of 0.5 m/s. Once T1, T2, and air velocity reached steady state, measurements were recorded for T1, T2, current, and voltage. The air velocity was increased in intervals of 0.5 m/s, and the steady state temperatures and air velocity were recorded. The final data was recorded when the maximum air velocity was attained with the damper fully open. 7 Progress Report Spring 2015 3.0 Resultsβ 0.65 From the data collected without the 100 °C, the calculated fractional heat loss due to natural convection (Qc/Qtot) decreased as the temperature of the heater surface (T2) increased. The fractional heat loss from radiant heat transfer (Qr/Qtot) increased as the temperature of the heater surface increased. Figures 3 and 4 show the results Qx/Qtot (x= c or r) blower on and at incremental temperatures of 0.6 0.55 0.5 Qr/Qto t 0.45 Qc/Qto t 0.4 0.35 0.3 0 200 400 600 T2 (K) Fig 3. Fractional Heat Loss versus Temperature of the heater surface (T2), Trial 1. from the two trials. When the blower was running and 0.65 temperature of the heater surface decreased as the effective air velocity (Ue) increased. Additionally, the rate of forced convective heat transfer (Qf) increased as the effective air velocity increased. Qx/Qtot (x=c or r) voltage and current were held constant, the 0.6 0.55 Qr/Qtot 0.5 0.45 Qc/Qtot 0.4 0.35 0.3 These results are 0 200 displayed in Figures 5 and 6, respectively. 400 600 T2 (K) Fig 4. Fractional Heat Loss versus Temperature of the heater surface (T2), Trial 2. 60 Qf (W) 50 40 30 Trial 1 Trial 2 20 10 Fig 5. Temperature of the heater surface (T2) versus Effective Air Velocity (Ue), Trials 1, 2. 0 5 10 15 Ue (m/s) Fig 6. Rate of Forced Convective Heat Transfer (Qf) versus Effective Air Velocity (Ue), Trials 1, 2 8 Progress Report Spring 2015 4.0 Analysis and Discussion of Resultsβ The first objective was to collect data on natural convection and radiation by measuring various temperatures, voltage and current. From these recorded values one can apply equations 2 and 4 in Appendix A-2 to calculate convective and radiant heat transfer. The total heat transfer can then be calculated by Equation 5 in Appendix A-2. Due to radiant and convective heat transfer a linear relationship is found when the temperature of the heater surface is graphed with the fractional heat transfer. As expected, when the fractional heat loss due to radiant heat transfer is large, at a specific temperature, the fractional heat transfer due to convective heat transfer is small. There is no other energy transfer happening, therefore the fractional heat transfer due to radiation must equal 1-(fractional heat transfer due to convection). This can be explained by the first law of thermodynamics, which states that energy cannot be created or destroyed but can change forms. As the temperature of the heater surface increases, the fractional heat transfer due to radiation increases and the fractional heat transfer due to convection decreases. Convection is a slower method of heat transfer compared to radiation. Radiation is inefficient but does not require material-to-fluid contact as convection does, therefore as the temperature increases the amount of radiation increases since radiation has a faster transfer time than convection. Objective two was obtained by varying the air velocity while the current and voltage stayed constant to investigate the effect of forced convection. As the air velocity increased, the temperature at thermocouple two decreased in both trials. This relationship allows for the conclusion that the surface cools as the air velocity is increased, the higher the air velocity the cooler the surface. As the air velocity increases past 8 m/s the temperature at the heater surface starts to plateau. One can make an inference that if the velocity could be increased past 10.5 m/s that the temperature at the heater surface would remain fairly constant at around 500K. As the air velocity increased the rate of forced convective heat transfer (Qf) increased, but the rate of increase of forced convective heat transfer slows as the air velocity is increased. As seen in Appendix A-2, Equation 16, the forced convective heat transfer is only dependent on the difference in temperature between the surface and the surrounding air. As the air moves faster through the cylinder, the temperature at the heater surface will decrease. While the temperature at the heater surface decreases, the temperature of the surrounding air 9 Progress Report Spring 2015 remains fairly constant. This phenomenon will result in a smaller slope as the velocity increases. Errors in results may have stemmed from the door to the laboratory being opened and shut as students left. When the door was opened the air velocity would fluctuate, thus making it difficult to record accurate values for velocity and temperature. Velocity also fluctuated when people walked by briskly. As the air velocity became constant the temperature fluctuated and vice versa. 5.0 Summary and Conclusionsβ The project involved the investigation of natural convection and radiation as well as forced convection. For Objective 1, the goal was to determine the fraction of total heat loss that occurs by radiation and natural convection to the air at steady-state. The difference between natural and forced convection is that natural convection does not utilize external forces such as, in the case of this experimentβs equipment, an air blower. Natural convection is caused by forces from density differences caused by temperature (i.e. warmer air moving upward and cooler air moving downward) variations in the air. Therefore, Objective 1 required that the damper be fully opened and the air blower remain completely off. To measure the variation of radiant heat transfer, the surface temperature of the equipment was increased. Two trials on two separate occasions were conducted and the results indicated the fractional heat loss from natural convection (QC/Qtotal) decreased as the temperature increased the heat loss. Ranging from 0.6 down to 0.39 while the temperature ranged from 100 K to 400 K. Additionally the fractional heat loss from radiant heat transfer increased as the temperature increased. This is a logical conclusion because of radiant heatβs direct correlation with temperature to the fourth power. The results from both trials were consistent with one another. The conclusion can be made that the proportion of heat lost at relatively low cylinder temperatures is predominantly due to convection and the proportion of heat lost at raised cylinder temperatures is lost from the surface due to radiation. Technical Objective 2 involved the evaluation of the effect of air velocity on the heating element surface temperature. A change had to be made to the steadied velocity of the damper which, in Objective 1, was set to 0.5 m/s and increased at intervals of additional an 0.5 m/s throughout the objective to completion. Forced convective heat transfer directly 10 Progress Report Spring 2015 correlates to velocity because the motion of the fluid increases heat transfer. The air velocity is controlled using the damper, as air velocity increased the rate of heat transfer also increased. In support of this statement the results indicated that when the blower was running at steady state (constant voltage and current), the air velocity increased as did the rate of forced convective heat transfer by the air blower. 6.0 Future Work β To further the work and fine tune the results of this particular experiment the changes that would need to be made are few. The results from Trials 1 and 2 were consistent with one another and with the samples in the lab book. In terms of error the biggest source of it during this experiment was the inconsistencies in air flow when other students would walk past the apparatus or the lab door would open and shut. Improvements can be made simply by moving the equipment to an area with lower human traffic or conducting the experiment during a time when the lab is not so crowded. The movement of external forces that were not caused by the air blower made the recording of data more time consuming because it was more difficult to get cylindrical duct values steady at the same time as the heating transfer unit. 7.0 Referencesβ [1] βMethods of Heat Transfer.β (2015). The Physics Classroom. http://www.physicsclassroom.com/class/thermalP/Lesson-1/Methods-of-Heat-Transfer [2] M. Bahrami. βForced Convection Heat Transfer.β Simon Fraser University. http://www.sfu.ca/~mbahrami/ENSC%20388/Notes/Forced%20Convection.pdf 11 Progress Report Spring 2015 Appendix A-1 Table 1. Trial 1: Data to determine the fractional heat loss from radiant heat transfer and convective heat transfer. T1 T1 T1 T2 T2 T2 I I I (°C) (°C) (°C) (°C) (°C) (°C) (Amps (Amps) (Amps V Low V High V Ave. Low High Ave. Low High Ave. ) Low High ) Ave. (Volts) (Volts) (Volts) 21.0 21.3 21.0 21.3 21.0 21.3 99.6 202.5 99.6 202.6 99.6 202.5 0.072 0.134 0.074 0.137 0.073 0.135 49 84 50 97 49.5 90.5 21.7 22.0 21.7 22.0 21.7 22.0 302.4 397.5 302.5 397.6 302.4 397.5 0.179 0.23 0.184 0.233 0.181 0.231 113 143 115 145 114 144 Table 2. Trial 1. Radiant and Convective Heat Transfer Coefficients, Radiant and Conductive Heat Transfer, Total Heat Transfer, Input Heat, and Fractional Heat Loss due to radiation and convection. T1 (°C) Ave. T2 (°C) Ave. hr (W/m^2 K) Qr (W) hc (W/m^2 K) Qc (W) Qtot (W) QInput (W) Qr/Qtot Qc/Qtot 21.0 99.60 8.52 1.47 12.50 2.16 3.63 3.61 0.405 0.594 21.3 202.55 13.66 5.44 15.35 6.12 11.57 12.26 0.470 0.529 22.7 302.45 20.64 12.74 17.11 10.57 23.32 20.29 0.546 0.453 22.0 397.55 29.40 24.29 18.39 15.20 39.49 33.33 0.615 0.384 Table 3. Trial 2. Data to determine the fractional heat loss from radiant heat transfer and convective heat transfer T1 T1 T1 T2 T2 T2 (°C) (°C) (°C) (°C) (°C) (°C) I(Amp I(Amp I(Amp V Low V High V Ave. Low High Ave. Low High Ave. s) Low s) High s) Ave. (Volts) (Volts) (Volts) 21.6 21.8 22.3 22.4 21.6 21.8 22.3 22.4 21.6 21.8 22.3 22.4 100.1 203.4 300.1 400.4 100.2 204.4 300.5 400.6 100.15 203.9 300.3 400.5 0.071 0.136 0.186 0.235 0.074 0.139 0.187 0.239 0.0725 0.1375 0.1865 0.237 51 89 115 147 52 90 116 149 51.5 89.5 115.5 148 Table 4. Trial 2. Radiant and Convective Heat Transfer Coefficients, Radiant and Conductive Heat Transfer, Total Heat Transfer, Input Heat, and Fractional Heat Loss due to radiation and convection. T1 (°C) T2 (°C) hr hc Ave. Ave. (W/m^2 (W/m^2 QInput K) Qr (W) K) Qc (W) Qtot (W) (W) Qr/Qtot Qc/Qtot 21.6 100.15 8.56 1.48 12.50 2.16 3.64 3.73 0.406 0.593 21.8 203.90 13.76 5.51 15.37 6.15 11.67 12.30 0.472 0.527 22.3 300.30 20.50 12.53 17.07 10.44 22.98 21.54 0.545 0.454 22.4 400.50 29.73 24.73 18.43 15.33 12 40.068 35.07 0.617 0.382 Progress Report Spring 2015 Table 5. Trial 1. Data collected to Calculate Effect of Air Velocity on Temperature and Convective Heat Transfer T1 (°C) Ave. T2 (°C) Ave. 24.8 25.7 26.3 26.6 26.9 26.9 26.9 26.9 26.8 26.6 375.25 365.30 355.90 336.05 321.45 313.75 306.45 298.45 290.70 274.25 Ua (m/s) Ave. 0.51 1.01 1.47 2.00 2.52 3.07 3.53 4.07 4.44 4.91 I (Amps) Ave. 0.327 0.332 0.332 0.330 0.331 0.333 0.334 0.332 0.334 0.333 V (Volts) 200.0 201.5 200.0 200.0 200.5 200.0 202.5 202.0 201.5 200.5 26.3 26.1 263.55 247.45 5.42 5.90 0.335 0.329 202.5 200.0 25.9 25.8 236.65 230.15 6.58 7.00 0.331 0.330 199.5 199.5 25.6 25.8 25.8 25.7 25.8 224.05 222.45 220.45 220.05 221.25 7.47 8.16 8.48 9.18 9.48 0.329 0.329 0.331 0.330 0.333 198.5 198.5 198.5 198.5 199.5 Table 6. Trial 1. Calculated Variables for the Calculation of Forced Convective Heat Transfer. T1 (K) T2 (K) Ue (m/s) v(m^2/s) Pr k (W/mK) Re Nu 297.8 298.7 299.3 299.6 648.25 638.30 628.90 609.05 0.62 1.23 1.79 2.44 1.5462E-05 1.5548E-05 1.5609E-05 1.5639E-05 0.708 0.708 0.708 0.708 0.0260 0.0261 0.0261 0.0262 402.41 792.51 1152.86 1564.06 10.40 14.69 17.84 20.93 299.9 299.9 299.9 299.9 299.8 299.6 299.3 299.1 298.9 298.8 594.45 586.75 579.45 571.45 563.70 547.25 536.55 520.45 509.65 503.15 3.08 3.75 4.31 4.96 5.41 5.99 6.61 7.19 8.03 8.54 1.567E-05 1.567E-05 1.567E-05 1.567E-05 1.566E-05 1.5639E-05 1.5614E-05 1.5589E-05 1.5568E-05 1.5558E-05 0.708 0.708 0.708 0.708 0.708 0.708 0.708 0.708 0.708 0.708 0.0262 0.0262 0.0262 0.0262 0.0262 0.0262 0.0261 0.0261 0.0261 0.0261 1965.87 2394.08 2752.22 3168.76 3459.07 3830.20 4234.92 4617.48 5160.31 5493.02 23.62 26.24 28.28 30.52 32.01 33.85 35.76 37.51 39.89 41.31 298.6 298.8 497.05 495.45 9.11 9.95 1.5538E-05 1.5558E-05 0.708 0.708 0.0261 0.0261 5869.25 6398.73 42.86 44.99 298.8 298.7 298.8 493.45 493.05 494.25 10.34 11.19 11.25 1.5558E-05 1.5548E-05 1.5558E-05 0.708 0.708 0.708 0.0261 0.0261 0.0261 6649.66 7203.27 7402.45 45.97 48.09 48.84 13 Progress Report Spring 2015 Table 7. Trial 1. Temperatures of Surrounding Air (T1) and Heater Surface (T2), Effective Air Velocities, Forced Convective Heat Transfer Coefficient, and Forced Convective Heat Transfer. T1 (K) T2 (K) Ue (m/s) hf (W/M^2*K) Qf (W) 297.8 648.25 0.62 27.14 20.92 298.7 638.30 1.23 38.41 28.69 299.3 628.90 1.79 46.71 33.87 299.6 609.05 2.44 54.86 37.34 299.9 594.45 3.08 61.97 40.15 299.9 586.75 3.75 68.84 43.44 299.9 579.45 4.31 74.20 45.63 299.9 571.45 4.96 80.07 47.84 299.8 563.70 5.41 83.96 48.74 299.6 547.25 5.99 88.72 48.33 299.3 536.55 6.61 93.68 48.88 299.1 520.45 7.19 98.18 47.81 298.9 509.65 8.03 104.36 48.38 298.8 503.15 8.54 108.02 48.56 298.6 497.05 9.11 112.03 48.91 298.8 495.45 9.95 117.65 50.89 298.8 493.45 10.34 120.22 51.48 298.7 493.05 11.19 125.73 53.76 298.8 494.25 11.25 127.72 54.92 Table 8. Trial 2. Data collected to Calculate Effect of Air Velocity on Temperature and Convective Heat Transfer T1 (°C) Ave. T2 (°C) Ave. Ua (m/s) Ave. I (Amps) Ave. 24.3 25.5 26.3 27.0 378.4 373.3 355.7 329.8 0.45 1.00 1.49 2.09 0.328 0.331 0.331 0.330 199.5 202.0 202.5 201.5 27.1 27.3 27.1 26.8 27.0 26.7 26.4 26.2 25.9 25.7 322.4 311.7 300.0 294.1 278.7 274.3 256.1 244.9 232.9 229.5 2.45 2.93 3.45 4.10 4.48 4.91 5.50 5.91 6.55 6.87 0.330 0.331 0.329 0.331 0.329 0.330 0.331 0.330 0.330 0.332 199.5 200.5 199.5 200.0 199.5 200.0 200.5 199.5 199.5 199.5 25.6 25.7 25.7 25.6 227.0 223.7 223.0 220.9 7.60 8.00 8.54 9.00 0.333 0.329 0.329 0.329 199.0 199.5 197.5 198.5 14 V Ave. (Volts) Progress Report Spring 2015 Table 9. Trial 2. Calculated Variables for the Calculation of Forced Convective Heat Transfer. T1 (K) T2 (K) v (m^2/s) Pr k (W/mK) Re Nu 297.3 298.5 299.3 300.0 300.1 300.3 300.1 299.8 300.0 299.7 651.45 646.35 628.75 602.85 595.45 584.75 573.05 567.15 551.75 547.35 1.54057E-05 1.55276E-05 1.56089E-05 0.00001568 1.56902E-05 1.57105E-05 1.56902E-05 1.56597E-05 0.00001568 1.56495E-05 0.708 0.708 0.708 0.708 0.707 0.707 0.707 0.708 0.708 0.708 0.0260 0.0261 0.0261 0.0262 0.0262 0.0262 0.0262 0.0262 0.0262 0.0262 360.32 789.62 1164.59 1630.03 1908.90 2279.17 2682.57 3198.08 3485.71 3827.72 9.84 14.66 17.93 21.39 23.25 25.56 27.89 30.68 32.15 33.83 299.4 299.2 529.10 517.90 1.5619E-05 1.55987E-05 0.708 0.708 0.0261 0.0261 4299.94 4622.30 36.06 37.53 298.9 298.7 505.95 502.50 1.55682E-05 1.55479E-05 0.708 0.708 0.0261 0.0261 5132.88 5390.68 39.78 40.88 298.6 298.7 298.7 298.6 500.00 496.75 496.05 493.95 1.55378E-05 1.55479E-05 1.55479E-05 1.55378E-05 0.708 0.708 0.708 0.708 0.0261 0.0261 0.0261 0.0261 5967.39 6281.29 6705.01 7066.65 43.26 44.52 46.19 47.58 Table 10. Trial 2. Temperatures of Surrounding Air (T1) and Heater Surface (T2), Effective Air Velocities, Forced Convective Heat Transfer Coefficient, and Forced Convective Heat Transfer. T1 (K) T2 (K) Ue (m/s) hf (W/M^2*K) Qf (W) 297.3 298.5 299.3 300.0 300.1 300.3 300.1 299.8 300.0 651.45 646.35 628.75 602.85 595.45 584.75 573.05 567.15 551.75 0.55 1.22 1.81 2.55 2.99 3.58 4.20 5.00 5.46 25.64 38.319 46.96 56.13 61.04 67.12 73.22 80.45 84.36 19.97 29.32 34.03 37.39 39.66 42.00 43.97 47.32 46.72 299.7 299.4 299.2 298.9 298.7 298.6 547.35 529.1 517.9 505.95 502.5 500.0 5.99 6.71 7.21 7.99 8.38 9.27 88.71 94.48 98.27 104.05 106.86 113.07 48.33 47.74 47.28 47.39 47.91 50.10 298.7 298.7 298.6 496.75 496.05 493.95 9.76 10.42 10.98 116.40 120.75 124.34 50.71 52.42 53.44 15 Progress Report Spring 2015 Appendix A-2: Example Calculations Constant Data π = ππππ π ππ£ππ‘π¦ = 1.0 πΉ = πβπππ πΉπππ‘ππ = 1.0 π = ππ‘ππππ π΅πππ‘π§πππ ππππ π‘πππ‘ = 5.67 β 10β8 π/ππΎ π΄π = π»πππ‘ππ ππ’πππππ π΄πππ ππ π‘βπ πΆπ¦ππππππ = 0.0022 π2 D= diameter of cylinder= .01 m Calculation for Radiant Heat Transfer Coefficient βπ = π β πΉ β π (π24 β π14 )/(π2 β π1 ) βπ = 1.0 β 1.0 β 1.0 (1) 372.754 β 294.154 π = 8.52 2 372.75 β 294.15 π πΎ Calculation for Radiant Heat Transfer ππ= βπ β π΄π β (π2 β π1 ) (2) ππ= 8.52 β .0022 β (372.75 β 294.15) = 1.47 π Calculation for Convective Heat Transfer Coefficient βπ = 1.32 ( βπ = 1.32 ( π2βπ1 .25 π· ) 372.15 β 294.15 ) . 01 (3) .25 = 12.50 π π2 πΎ Calculation for Convective Heat Transfer ππ = βπ β π΄π β (π2 β π1 ) (4) ππ = 12.50 β .0022 β (372.15 β 294.15) = 2.16 π Calculation for Total Heat Transfer ππ‘ππ‘ = ππ + πβ ππ‘ππ‘ = 2.16 + 1.27 = 3.63 π 16 (5) Progress Report Spring 2015 Calculation for Fractional Heat Loss due Radiation πΉππππ‘πππππ π»πππ‘ πΏππ π ππ’π π‘π π πππππ‘πππ = ππ /ππ‘ππ‘ πΉππππ‘πππππ π»πππ‘ πΏππ π ππ’π π‘π π πππππ‘πππ = (6) 1.47 = 0.405 3.63 Calculation for Fractional Heat Loss due Natural Convection πΉππππ‘πππππ π»πππ‘ πΏππ π ππ’π π‘π πππ‘π’πππ πΆπππ£πππ‘πππ = ππ /ππ‘ππ‘ πΉππππ‘πππππ π»πππ‘ πΏππ π ππ’π π‘π πππ‘π’πππ πΆπππ£πππ‘πππ = (7) 2.16 = 0.594 3.63 Calculation for Effective Air Velocity ππ = 1.22 β π΄ππ πππππππ‘π¦ (8) ππ = 1.22 β .51 = .62 π/π Calculation for Dynamic Viscosity of Air π£ = (1.016 β 10β7 )π1 β 1.48 β 10β7 (9) π£ = (1.016 β 10β7 )297.8 β 1.48 β 10β7 = 1.546 β 10β5 π2 π Calculation for Prandtl Number ππ = (β2.2 β 10β4 )π1 + .774 (10) ππ = (β2.2 β 10β4 )297.8 + .774 = .708 Calculation for Thermal Conductivity Constant π = (7.58 β 10β5 )π1 + .0035 π = (7.58 β 10β5 )297.8 + .0035 = .0260 (11) π ππΎ Calculation for Reynolds Number π π = π π = ππ βπ· π£ . 62 β .01 = 402.41 1.546 β 10β5 17 (12) Progress Report Spring 2015 Calculation for Nusselt Number (.62βπ π .5 βππ .33 ) ππ’ = .3 + .25 .4 .66 ) ) ππ × (1 + ( (1+( ππ’ = .3 + (. 62 β 402.41.5 β. 708.33 ) . 4 .66 (1 + (. 708) ) .25 π π 282000 .5 ) ) (13) . 708 .5 × (1 + ( ) ) = 10.40 282000 Calculation for Forced Convective Heat Transfer Coefficient π βπ = π· β ππ’ βπ = (14) . 0260 π β 10.40 = 27.14 2 . 01 π πΎ Calculation for Forced Convective Heat Transfer ππ = βπ β π΄π β (π2 β π1 ) ππ= 27.14 β .0022 β (372.75 β 294.15) = 20.92 π 18 (15)